Shifted Chebyshev equation and polynomial termination ($z(z+2)y^{\prime \prime }+(z+1)y^{\prime }-{\alpha }^{2}y=0$)

At zero the indicial equation is $\sigma (2\sigma -1)=0$. The two Frobenius series have exponents zero and one half, with recurrenceThe denominators never vanish for these exponents. Unless a series terminates its coefficient ratio tends to $-1/2$, giving convergence within distance two of zero. If $\alpha =m$ is a positive integer, the exponent-zero series terminates at degree $m$. The substitution $t=z+1$ gives the standard Chebyshev differential equation, and this solution is a multiple of $T_m(z+1)$. For half-integer $\alpha$, termination can instead occur in the power-series factor of the exponent-one-half solution.